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I wanted to know why it isn't statistically correct or correct in a probabilistic sense to subtract the lower confidence limits and upper confidence limits of the separate confidence intervals of two population means to generate the confidence interval of the difference between the two means (or for e.g Difference between Two Population Proportions etc... ).

A. Napster
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    Essentially because the standard error of the difference is not the difference of the standard errors. – ocram May 11 '20 at 10:58
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    See https://stats.stackexchange.com/a/18259/919 for a discussion of what subtracting the confidence limits actually does. – whuber May 11 '20 at 11:46

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